Changing “We Tinies, We Toonies are..." to "We're Tinies, we're Toonies are..." would mean "We are Tinies are.." Not only does that not make sense in English, it’s not what is being said in German. “We Tinies, we Toonies, are shrill, fun Cartoonies” is a literal, word-for-word translation.

“We're Tiny Toon doctors recommended" also does not work because this would mean "we are recommend," which doesn't make sense in English and would be an incorrect translation.

In both instances, some additional information about the subject has been added before the verb comes along, which is perfectly acceptable in both languages. In fact, the first line of our US Constitution does this same thing: "We the people of the United States...."